Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622953 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#459 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1997 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,16)(3,14,20,32)(4,21,24,49)(5,12,25,38)(6,15,8,13)(7,10,19,33)(11,34,37,17)(18,40,45,56)(22,48,50,63)(23,52,30,42)(26,41,28,39)(27,31,46,55)(29,43,47,36)(35,57,58,44)(51,64,54,60)(53,61,62,59);(1,3,8,19,9,20,6,7)(2,10,15,32,16,33,13,14)(4,17,30,47,24,34,23,29)(5,18,28,46,25,45,26,27)(11,21,43,52,37,49,36,42)(12,31,41,56,38,55,39,40)(22,44,54,62,50,57,51,53)(35,48,61,64,58,63,59,60);(1,4)(2,11)(3,17)(5,22)(6,23)(7,29)(8,30)(9,24)(10,21)(12,35)(13,36)(14,42)(15,43)(16,37)(18,44)(19,47)(20,34)(25,50)(26,51)(27,53)(28,54)(31,48)(32,52)(33,49)(38,58)(39,59)(40,60)(41,61)(45,57)(46,62)(55,63)(56,64);(1,5)(2,12)(3,18)(4,22)(6,26)(7,27)(8,28)(9,25)(10,31)(11,35)(13,39)(14,40)(15,41)(16,38)(17,44)(19,46)(20,45)(21,48)(23,51)(24,50)(29,53)(30,54)(32,56)(33,55)(34,57)(36,59)(37,58)(42,60)(43,61)(47,62)(49,63)(52,64);(5,25)(12,38)(18,45)(22,50)(26,28)(27,46)(31,55)(35,58)(39,41)(40,56)(44,57)(48,63)(51,54)(53,62)(59,61)(60,64)Polycyclic presentation:
2^9|a^2=i,b^2=fi,f^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gh,h^a=hi,g^c=gh,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530151);